Question

cho a/b=c/d chứng minh rằng:

a)\(\dfrac{ab}{cd}=\dfrac{a^2+b^2}{c^2+d^2}\) b)\(\dfrac{ac}{bd}=\dfrac{a^2+c^2}{b^2+d^2}\)

Answer 1

Đặt:

\(\dfrac{a}{b}=\dfrac{c}{d}=t\Leftrightarrow\left\{{}\begin{matrix}a=bt\\c=dt\end{matrix}\right.\)

a) \(\left\{{}\begin{matrix}\dfrac{ab}{cd}=\dfrac{b^2t}{d^2t}=\dfrac{b^2}{d^2}\\\dfrac{a^2+b^2}{c^2+d^2}=\dfrac{b^2t^2+b^2}{d^2t^2+d^2}=\dfrac{b^2\left(t^2+1\right)}{d^2\left(t^2+1\right)}=\dfrac{b^2}{d^2}\end{matrix}\right.\Rightarrowđpcm\)

b)\(\left\{{}\begin{matrix}\dfrac{ac}{bd}=\dfrac{t^2bd}{bd}=t^2\\\dfrac{a^2+c^2}{b^2+d^2}=\dfrac{b^2t^2+d^2t^2}{b^2+d^2}=\dfrac{t^2\left(b^2+d^2\right)}{b^2+d^2}\end{matrix}\right.\Rightarrowđpcm\)